Classifying space: Difference between revisions
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Another way of saying this is that the classifying space of a discrete group is a path-connected space with the given discrete group as fundamental group, and whose universal cover is a [[weakly contractible space]] (often, a [[contractible space]]). | Another way of saying this is that the classifying space of a discrete group is a path-connected space with the given discrete group as fundamental group, and whose universal cover is a [[weakly contractible space]] (often, a [[contractible space]]). | ||
Classifying spaces for discrete groups are special cases of [[Eilenberg-Maclane space]]s. | |||
A topological space occurs as a classifying space for a discrete group iff it is [[aspherical space|aspherical]]. | |||
Latest revision as of 19:40, 11 May 2008
Definition
Let be a topological group. A classifying space of , denoted , is defined as the quotient of a weakly contractible space (a space all whose homotopy groups are trivial) by a free action of .
In particular when is a discrete group, a classifying space of , denoted , is a path-connected space whose fundamental group is and for which the higher homotopy groups are trivial.
Another way of saying this is that the classifying space of a discrete group is a path-connected space with the given discrete group as fundamental group, and whose universal cover is a weakly contractible space (often, a contractible space).
Classifying spaces for discrete groups are special cases of Eilenberg-Maclane spaces.
A topological space occurs as a classifying space for a discrete group iff it is aspherical.